This dissertation comprises four essays in financial economics. In the first essay, we conduct inference on volatility with noisy high-frequency data. The second essay proposes a semi- parametric approach to disentangling the autocovariance of equity returns at high frequency. In the third essay, we consider the limit of arbitrage arising from learning difficulty when in- vestors are facing a large number of investment opportunities. Finally, we investigate market efficiency in the presence of many investors.